Answer :
To classify the number, we follow these steps:
1. **Check for a Perfect Square:**
We look for an integer $n$ such that
$$ n^2 = 196. $$
Calculating the square root:
$$ \sqrt{196} = 14. $$
Since
$$ 14 \times 14 = 196, $$
the number $196$ is a perfect square.
2. **Check for a Perfect Cube:**
We look for an integer $m$ such that
$$ m^3 = 196. $$
Testing the values around the estimated cube root:
- $$ 5^3 = 125 $$
- $$ 6^3 = 216 $$
The number $196$ lies between $125$ and $216$, so there is no integer $m$ that satisfies $m^3 = 196$. Hence, $196$ is not a perfect cube.
3. **Conclusion:**
Since $196$ is a perfect square but not a perfect cube, the correct classification is that it is a perfect square.
Thus, the correct answer is:
$$\textbf{Option C: The number is a perfect square because } \square \times \square = 196.$$
1. **Check for a Perfect Square:**
We look for an integer $n$ such that
$$ n^2 = 196. $$
Calculating the square root:
$$ \sqrt{196} = 14. $$
Since
$$ 14 \times 14 = 196, $$
the number $196$ is a perfect square.
2. **Check for a Perfect Cube:**
We look for an integer $m$ such that
$$ m^3 = 196. $$
Testing the values around the estimated cube root:
- $$ 5^3 = 125 $$
- $$ 6^3 = 216 $$
The number $196$ lies between $125$ and $216$, so there is no integer $m$ that satisfies $m^3 = 196$. Hence, $196$ is not a perfect cube.
3. **Conclusion:**
Since $196$ is a perfect square but not a perfect cube, the correct classification is that it is a perfect square.
Thus, the correct answer is:
$$\textbf{Option C: The number is a perfect square because } \square \times \square = 196.$$
